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Multistep Monotone Recursive Methods
Leonard J. MirmanUniversity of Virginia
Kevin L. ReettArizona State University
December, 2003Preliminary and Incomplete
A powerful new class of monotone recursive methods is described forstudying the existence, characterization, and computation of recursivecompetitive equilibrium in a large class of innite horizon economies withcapital, production, public policy, behavioral heterogeneity, and incom-plete markets. The analysis combines aspects of topology in normal cones,lattice theory, and interval analysis. The methods are multistep in thesense that recursive equilibrium are computed using compositions of xedpoint operators. The class of economies studied covers many situationswhere traditional monotone map methods (e.g, Coleman  and relatedconstructions) can not be applied, and includes models such as the modelof Bewley  and Krusell and Smith . The methods are constructiveand combine aspects of topological and lattice theoretic xed point the-ory. Discussions of the implications for characterizing the accuracy of ofnumerical solutions to the class of economies is discussed, and some basiccomparative statics results are provided.
Since the original work of Lucas and Prescott , Brock and Mirman ,and Prescott and Mehra , recursive methods and dynamic programminghave become staples in the toolkit of many researchers in macroeconomics, eco-nomic growth, monetary economics and public nance.1 The methods have
The authors would like to thank Manjira Datta, Tom Krebs, Olivier Morand, MarioPascoa, Richard Rogerson, and Manuel Santos as well as the participants of the 2003 SAETConference in Rodos Greece for helpful discussions. Reett would like to thank the DeansAward in Excellence Summer Research Program at Arizona State for generously funding thisresearch. This is a very preliminary draft of the paper. Do not circulate the current draft,rather email the authors at Kevin.email@example.com for the most recent version.
1The use of recursive methods in economics dynamics has been extensive over the last thirtyyears. See for example, the monograph of Stokey, Lucas, with Prescott  and Ljungqvistand Sargent  for historical developments and further motivation for the use and importanceof recursive methods in studying economic dynamics.
been shown to provide sharp characterizations of recursive (Markovian) equilib-rium in many situations where the second welfare theorem is available. Theseresults have been obtained by applying results in the dynamic programmingliterature of the 1960s in conjunction with the second welfare theorem to pro-vide sharp characterizations of recursive versions of central planning problems.Further given recent developments of lattice programming methods for dynamiceconomies in the work of Amir, Mirman, and Perkins  and Mirman, Morand,and Reett , monotone comparative dynamics on the space of economiesfor a large class of Pareto optimal economies is also available on the entire setof recursive equilibrium. As numerical solution algorithms based the monotonecontraction mapping theorem for dynamic programming is often available, sharpcharacterizations of numerical implementations of these recursive methods arealso well-known (e.g, Santos and Vigo ). Therefore when the second welfaretheorem is available, it has been found that the theory, characterization, andcomputation of recursive competitive equilibrium can be unied to provide avery powerful systematic approach to the study of the structure of Markovianequilibrium in a wide array of Arrow-Debreu-McKenzie settings.Unfortunately, the development of a similar collection of recursive methods
powerful enough to characterize the set of Markovian equilibrium in environ-ments where the second welfare theorem fails has been much more elusive, es-pecially in settings where there is activist or passive public policy (either scalor monetary), incomplete asset markets, behavioral heterogeneity, and/or mul-tisector production. In homogeneous agent economies with complete marketsand equilibrium distortions such as taxes and production nonconvexities, an ex-tensive set of methods are available. In particular a series of papers over thelast fteen years has provided a powerful collection of monotone recursive meth-ods based on order-theoretic xed point theory has been developed. Beginningwith Lucas and Stokey , Bizer and Judd  Coleman , and continuingwith a series of recent developments presented in Greenwood and Human ,Coleman , Datta, Mirman, and Reett , Morand and Reett ,and Mirman, Morand, and Reett , a set of monotone recursive methodsthat integrate existence, characterization, and computation of recursive equi-librium has been delivered that unies the study of existence, characterization,comparative statics on the space of economies, and numerical approximation ofMarkovian equilibrium.2
Unfortunately, these methods have yet to be extended systematically to com-petitive settings with many agents.3 As many interesting situations studied in
2For some interesting recent nonexistence of Markovian equilibrium results, see Santos and Krebs 
3There are two notable exceptions to this statement. First, the methods in Mirman,Morand, and Reett  actually work for symmetric equilibrium in some multiagent modelswith incomplete markets and aggregate and idiosyncratic risk (namely models with a niteor countable number of ex post heterogeneity). More directly though, a recent paper byDatta, Mirman, Morand, and Reett , monotone map methods have been extended to thestudy of stationary Ramsey equilibrium problems with public policy (related to the work ofBecker and Zilcha ). In this paper, the authors develop order theoretic topological xedpoint methods based on monotone operators that compute Markovian equilibrium in the
applied work in macroeconomics, public nance, economic growth and develop-ment, and dynamic industrial organization often involve a role for public policy(scal or monetary), production nonconvexities, and incomplete markets, thisshortcoming has been a serious impediment to unifying theoretical and com-putation approaches to the decentralization of these economies, and the char-acterization of recursive numerical algorithms attempting tied to theoreticalconstructions to construct Markovian equilibrium via successive approximationis essentially nonexistent.4
This paper takes a large rst step in bridging the gap between theoreticaland numerical implementations of monotone recursive methods. It proposes anew collection of xed point algorithms capable of unifying topological aspectsof the problem with recent lattice theoretic developments. We then apply themethods to a collection of economies where traditional Monotone Map methodssuch as Coleman  are not apparently available. To distinguish our meth-ods from those approaches in the current literature, we rst present proofs ofexistence of recursive equilibrium based on nonconstructive topological xedpoint arguments using a version of Schauders theorem. In some sense, whileour methods are reminiscent of some existing approaches to the problem in theliterature using local convexity arguments (e.g, Miao ), they are also quitedierent. In particular, what is interesting though is that unlike many currentapproaches (e.g, Bergin and Bernhardt  and Miao ), we approachto the existence question by studying xed point constructions on the spaceof candidate equilibrium "policy functions" as opposed to sequences of equilib-rium distributions. This allows us to actually deliver new characterizations ofrecursive equilibrium that are not provided in this recent existing body of work.To avoid a complete reliance on nonconstructive topological methods, our
methods exploit the presence of dynamic complementarities in the underlyingagents decision environment, but in general study these complementarities arean expanded state space. The resulting "multistep" monotone method there-fore in the end is based on an equilibrium version of household Euler equationinequality. The algorithm sequentially computes recursive equilibrium as fol-lows: (i) compute individual decision rules given the aggregate laws of motionfor the aggregate variables via a version of Tarskis theorem using an "ascend-ing" operator; (ii) exploit well-known comparative statics results for xed pointcorrespondences in a parameter in conjunction with additional topological reg-
spirit of those consider in the present paper. For these economies, it is shown that successiveapproximation schemes appear available.This extends the results in an earlier paper of Beckerand Zilcha  by providing the rst set of constructive xed point result on this problem, asopposed to appealing to local convexity arguments.As this paper will discuss, those methods do not appear available for the Bewley-style
models considered in papers such as Aiyagari , Huggett , Krusell and Smith , andMiao .
4There has been much work on methods to provide numerical solutions to dynamic equi-librium models (See for example, Judd  for an excellent summary of these methods).Unfortunately, formal arguments concerning the accuracy of these methods have not beenforthcoming (in part because theoretical approaches to studying the existence questions havebeen nonconstructive). For one line of work attempting to rectify this situation for nonoptimaleconomies, see the discussion in Mirman, Morand and Reett .
ularity conditions to dene a second stage operator that can be proven to be acompact antitone operator that are self-maps on a compact subsets of a normalcone )and therefore have a nonempty set of xed points), and nally (iii) developa